X-ray Tomography

Frank Natterer

Institut fur Numerische und Angewandte Mathematik, University of Munster,Einsteinstraße 62, 48149 Munster, Germanynatterer@math.uni-muenster.de

Summary. We give a survey on the mathematics of computerized tomography. Westart with a short introduction to integral geometry, concentrating on inversion formu-las, stability, and ranges. We then go over to inversion algorithms. We give a detailedanalysis of the filtered backprojection algorithm in the light of the sampling theorem.We also describe the convergence properties of iterative algorithms. We shortly mentionFourier based algorithms and the recent progresses made in their accurate implemen-tation. We conclude with the basics of algorithms for cone beam scanning which is thestandard scanning mode in present days clinical practice.

1 Introduction

X-ray tomography, or computerized tomography (CT) is a technique for imagingcross sections or slices (slice = τoµos (greek)) of the human body. It was intro-duced in clinical practice in the 70s of the last century and has revolutionizedclinical radiology. See Webb (1990), Natterer and Ritman (2002) for the historyof CT.

The mathematical problem behind CT is the reconstruction of a function f inR

2 from the set of its line integrals. This is a special case of integral geometry, i.e.the reconstruction of a function from integrals over lower dimensional manifolds.Thus, integral geometry is the backbone of the mathematical theory of CT.

CT was soon followed by other imaging modalities, such as emission tomo-graphy (single particle emission tomography (SPECT) and positron emissiontomography (PET)), magnetic resonance imaging (MRI). CT also found appli-cations in various branches of science and technology, e.g. in seismics, radar,electron microscopy, and flow. Standard references are Herman (1980), Kak andSlaney (1987), Natterer and Wubbeling (2001).

In these lectures we give an introduction into the mathematical theory andthe reconstruction algorithms of CT. We also discuss matters of resolution andstability. Necessary prerequisites are calculus in R

n, Fourier transforms, and theelements of sampling theory.

18 F. Natterer

Since the advent of CT many imaging techniques have come into being, someof them being only remotely related to the straight line paradigm of CT, suchas impedance tomography, optical tomography, and ultrasound transmission to-mography. For the study of these advanced technique a thorough understandingof CT is at least useful. Many of the fundamental tools and issues of CT, such asbackprojection, sampling, and high frequency analysis, have their counterpartsin the more advanced techniques and are most easily in the framework of CT.

The outline of the paper is as follows. We start with a short account of therelevant parts of integral geometry, with the Radon transform as the central tool.Then we discuss in some detail reconstruction algorithms, in particular the nec-essary discretizations for methods based on inversion formulas and convergenceproperties of iterative algorithm. Finally we concentrate on the 3D case whichis the subject of current research.

2 Integral Geometry

In this section we introduce the relevant integral transforms, derive inversionformulas, and study the ranges. For a thorough treatment see Gelfand, Graev,and Vilenkin (1965), Helgason (1999), Natterer (1986).

2.1 The Radon Transform

Let f be a function in Rn. To avoid purely technical difficulties we assume f to

be smooth and of compact support, if not said otherwise.For θ ∈ Sn−1, s ∈ R

1 we define

(Rf)(θ, s) =∫

x·θ=s

f(x)dx . (1)

R is the Radon transform. dx is the restriction of the Lebesgue measure in Rn

to x · θ = s. Other notations are

(Rf)(θ, s) =∫

θ⊥

f(sθ + y)dy (2)

with θ⊥ the subspace of Rn orthogonal to θ, and

(Rf)(θ, s) =∫

δ(x · θ − s)f(x)dx (3)

with δ the Dirac δ-function.The Radon transform is closely related to the Fourier transform

f(ξ) = (2π)−n/2

∫

Rn

e−ix·ξf(x)dx .

Namely, we have the so-called projection-slice theorem.

X-ray Tomography 19

Theorem 2.1.(Rf)∧(θ, σ) = (2π)(n−1)/2f(σθ) .

Note that the Fourier transform on the left hand side is the 1D Fouriertransform of Rf with respect to its second variable, while the Fourier transformon the right hand side is the nD Fourier transform of f .

For the proof of Theorem 2.1 we make use of the definition (2) of Radontransform, yielding

(Rf)∧(θ, σ) = (2π)−1/2

∫

R1

e−isσ(Rf)(θ, s)ds

= (2π)−1/2

∫

R1

e−isσ

∫

θ⊥

f(sθ + y)dyds .

Putting x = sθ + y, i.e. s = x · θ, we have

(Rf)∧(θ, s) = (2π)−1/2

∫

Rn

e−iσx·θf(x)dx

= (2π)(n−1)/2f(σθ) .

The proofs of many of the following results follow a similar pattern. So we omitproofs unless more sophisticated tools are needed.

It is possible to extend R to a bounded operator R : L2(Rn) → L2(Sn−1×R1).

As such R has an adjoint R∗ : L2(Sn−1 × R1) → L2(Rn) which is easily seen

to be(R∗g)(x) =

∫

Sn−1

g(θ, x · θ)dθ . (4)

R∗ is called the backprojection operator in the imaging literature.We also will make use of the Hilbert transform

(Hf)(s) =1π

∫f(t)s − t

dt (5)

which, in Fourier domain, is given by

(Hf)∧(σ) = −i sgn(σ)f(σ) (6)

with sgn(σ) the sign of the real number σ. Now we are ready to state and toprove Radon’s inversion formula:

Theorem 2.2. Let g = Rf . Then

f =12(2π)1−nR∗Hn−1g(n−1)

where g(n−1) stands for the derivative of order n − 1 of g with respect to thesecond argument.

20 F. Natterer

For the proof we write the Fourier inversion formula in polar coordinates andmake use of Theorem 2.1 and the evenness of g, i.e. g(θ, s) = g(−θ,−s).

Special cases of Theorem 2.2 are

f(x) =1

4π2

∫

S1

∫

R1

g′(θ, s)x · θ − s

dsdθ (7)

for n = 2 and

f(x) = − 18π2

∫

S2

g′′(θ, x · θ)dθ (8)

for n = 3. Note that there is a distinctive difference between (7) and (8): Thelatter one is local in the sense that for the reconstruction of f at x0 only theintegrals g(θ, s) over those planes x · θ = s are needed that pass through x0 ornearby. In contrast, (7) is not local in this sense, due to the integral over R1.

In order to study the stability of the inversion process we introduce theSobolev spaces Hα on R

n, Sn−1 × R1 with norms

‖f‖2Hα(Rn)

=∫

Rn

(1 + |ξ|2)α|f(ξ)|2dξ ,

‖g‖2

Hα(Sn−1×R1)=

∫

Sn−1

∫

R1

(1 + σ2)α|g(θ, σ)|2dσdθ .

As in Theorem 2.1, the Fourier transform g is the 1D Fourier transform of gwith respect to the second argument.

Theorem 2.3. There exist constants c, C > 0 depending only on α, n, such that

c‖f‖Hα(Rn) ≤ ‖Rf‖

Hα+(n−1)/2 (Sn−1×R1) ≤ C‖f‖Hα(Rn)

for f ∈ Hα(Rn) with support in |x| < 1.

The conclusion is that Rf is smoother than f by the order (n − 1)/2. Thisindicates that the reconstruction process, i.e. the inversion of R, is slightly ill-posed.

Finally we study the range of R.

Theorem 2.4. For m ≥ 0 an integer let

pm(θ) =∫

R1

sm(Rf)(θ, s)ds .

Then, pm is a homogeneous polynomial of degree m in θ.

The proof is by verification. The question wether or not the condition ofTheorem 2.4 is sufficient for a function g of θ, s being in the range of R is thesubject of the famous Helgason–Ludwig theorem.

Theorem 2.4 has applications to cases in which the data function g is notfully specified.

X-ray Tomography 21

2.2 The Ray Transform

For f a function in Rn, θ ∈ Sn−1, x ⊥ θ⊥ we define

(Pf)(θ, x) =∫

R1

f(x + tθ)dt (9)

P is called the ray (or X-ray) transform. The projection-slice theorem for P reads

Theorem 2.5.(Pf)∧(θ, ξ) = (2π)1/2f(ξ), ξ ∈ θ⊥ .

The Fourier transform on the left hand side is the (n − 1)D Fourier transformin θ⊥, while the Fourier transform on the right hand side is the nD Fouriertransform in R

n.

The adjoint ray transform, which we call backprojection again, is given by

(P ∗g)(x) =∫

Sn−1

g(θ,Eθx)dθ (10)

with Eθ the orthogonal projection onto θ⊥, i.e. Eθx = x− (x · θ)θ. We have thefollowing analogue of Radon’s inversion formula:

Theorem 2.6. Let g = Pf . Then

f =1

2π|Sn−1| P ∗I−1g

with the Riesz potential(I−1g)∧(ξ) = |ξ|g(ξ)

in θ⊥.

Theorem 2.6 is not as useful as Theorem 2.2. The reason is that for n ≥ 3 (forn = 2 Theorems 2.2 and 2.6 are equivalent) the dimension 2(n − 1) of the datafunction g is greater than the dimension n of f . Hence the problem of invertingP is vastly overdetermined. A useful formula would compute f from the valuesg(θ, ·) where θ is restricted to some set S0 ⊆ Sn−1. Under which conditions onS0 is f uniquely determined?

Theorem 2.7. Let n = 3, and assume that each great circle on S2 meets S0.Then, f is uniquely determined by g(θ, ·), θ ∈ S0.

The condition on S0 in Theorem 2.7 is called Orlov’s completeness condition.In the light of Theorem 2.5, the proof of Theorem 2.7 is almost trivial: Letξ ∈ R

3 \ 0. According to Orlov’s condition there exists θ ∈ S0 on the greatcircle ξ⊥ ∩ S2. For this θ, ξ ∈ θ⊥, hence

f(ξ) = (2π)−1/2g(θ, ξ)

by Theorem 2.5. Thus f is uniquely (and stably) determined by g on S0.

22 F. Natterer

An obvious example for a set S0 ⊆ S2 satisfying Orlov’s completeness con-dition is a great circle. In that case inversion of P is simply 2D Radon inversionon planes parallel to S0. Orlov gave an inversion formula for S0 a spherical zone.

2.3 The Cone-Beam Transform

The relevant integral transform for fully 3D X-ray tomography is the cone beamtransform in R

3

(Cf)(a, θ) =

∞∫

0

f(a + tθ)dt (11)

where θ ∈ S2. The main difference to the ray transform P is that the sourcepoint a is restricted to a source curve A surrounding the object to be imaged.

Of course the following question arises: Which condition on A guaranteesthat f is uniquely determined by (Cf)(a, ·)?

A first answer is: Whenever A has an accumulation point outside supp (f)(provided f is sufficiently regular). Unfortunately, inversion under this assump-tions is based on analytic continuation, a process that is notoriously unstable.So this result is useless for practical reconstruction.

In order to find a stable reconstruction method (for suitable source curves A)we make use of a relationship between C and R discovered by Grangeat (1991):

Theorem 2.8.

∂

∂s(Rf)(θ, s) |s=a·θ =

∫

θ⊥∩S2

∂

∂θ(Cf)(a, ω)dω .

The notation on the right hand side needs explication: (Cf)(a, ·) is a functionon S2. θ is a tangent vector to S2 for each ω ∈ S2 ∩ θ⊥. Thus the derivative ∂

∂θmakes sense on S2 ∩ θ⊥.

Since Theorem 2.8 is the basis for present day’s work on 3D reconstructionwe sketch two proofs. The first one is completely elementary. Obviously it sufficesto put θ = e3, the third unit vector, in which case it reads

∂

∂s(Rf)(θ, a3) =

∫

ω∈S1

⎡⎣ ∂

∂z(Cf)

⎛⎝a,

⎛⎝ω

z

⎞⎠

⎞⎠

⎤⎦

z=0

dω

where a3 is the third component of the source point a. It is easy to verify thatright- and left-hand side both coincide with

∫

R2

∂f

∂x3

⎛⎝a +

⎛⎝x′

0

⎞⎠

⎞⎠ dx′ .

The second proof makes use of the formula

X-ray Tomography 23

∫

Sn−1

(Cf)(a, ω)h(θ · ω)dω =∫

R1

(Rf)(θ, s)h(s − a · θ)ds

that holds – for suitable functions f in R2 – for h a function in R

1 homogeneousof degree 1 − n; see Hamaker et al. (1980).

Putting h = δ′ yields Theorem 2.8.

Theorem 2.9. Let the source curve A satisfy the following condition: Each planemeeting supp (f) intersects A transversally. Then, f is uniquely (and stably)determined by (Cf)(a, θ), a ∈ A, θ ∈ S2.

The condition on A in Theorem 2.9 is called the Kirillov–Tuy condition.The proof of Theorem 2.9 starts out from Radon’s inversion formula (8) for

n = 3:

f(x) = − 18π2

∫

S2

[∂2

∂s2Rf(θ, s)

]s=x·θ

dθ (12)

Now assume A satisfies the Kirillov–Tuy condition, and let x · θ = s be a planehitting supp (f). Then there exists a ∈ A such that a · θ = s, and

[∂

∂sRf(θ, s)

]s=x·θ

=[

∂

∂sRf(θ, s)

]s=a·θ

=∫

θ⊥∩S2

∂

∂θ(Cf)(a, ω)dω

is known by Theorem 2.8. Since A intersects the plane x · θ = s transversallythis applies also to the second derivative in (12). Hence the integral in (12) canbe computed from the known values of Cf for all planes x · θ = s hitting supp(f), and for the others it is zero.

Theorem 2.8 in conjunction with Radon’s inversion formula (8) is the basisfor reconstruction formulas for C with source curves A satisfying the Kirillov–Tuy condition. The simplest source curve one can think of, a circle around theobject, does not satisfy the Kirillov–Tuy condition. For a circular source curvean approximate inversion formula, the FDK formula, exists; see Feldkamp et al.(1984). In medical applications the source curve is a helix.

2.4 The Attenuated Radon Transform

Let n = 2. The attenuated Radon transform of f is defined to be

(Rµf)(θ, s) =∫

x·θ=s

e−(Cµ)(x,θ⊥)f(x)dx

24 F. Natterer

where µ is another function in R2 and θ⊥ is the unit vector perpendicular to

θ such that det (θ, θ⊥) = 1. This transform comes up in SPECT, where f the(sought-for) activity distribution and µ the attenuation map.

Rµ admits an inversion formula very similar to Radon’s inversion formulafor R:

Theorem 2.10. Let g = Rµf . Then,

f =14π

Re div R∗−µ(θe−hHehg)

where

h =12(I + iH)Rµ ,

(R∗µg)(x) =

∫

S1

e−(Cµ)(x,θ⊥)g(θ, x · θ)dθ .

This is Novikov’s inversion formula; see Novikov (2000). Note that the back-projection R∗

−µ is applied to a vector valued function. For µ = 0 Novikov’sformula reduces to Radon’s inversion formula (7).

There is also an extension of Theorem 2.4 to the attenuated case:

Theorem 2.11. Let k > m ≥ 0 integer and h the function from Theorem 2.10.Then, ∫

R1

∫

S1

smeikϕ+h(θ,s)(Rµf)(θ, s)dθds = 0

where θ =

⎛⎝ cos ϕ

sin ϕ

⎞⎠.

The proofs of Theorems 2.10 and 2.11 are quite involved. They are bothbased on the following remark: Let

u(x, θ) = h(θ, x · θ) − (Cµ)(x, θ⊥) .

Then, u admits a Fourier series representation as

u(x, θ) =∑

>0 odd

u(x)eiϕ

with certain functions u(x) and θ =

⎛⎝ cos ϕ

sinϕ

⎞⎠.

Of course this amounts to saying that u(x, ·) admits an analytic continuationfrom S1 into the unit disk.

X-ray Tomography 25

2.5 Vectorial Transforms

Now let f be a vector field. The scalar transforms P , R can be applied compo-nentwise, so Pf , Rf make sense. They give rise to the vectorial ray transform

(Pf)(θ, s) = θ · (Pf)(θ, s) , x ∈ θ⊥

and the Radon normal transform

(R⊥f)(θ, s) = θ · (Rf)(θ, s) .

Obviously, vectorial transforms can’t be invertible. In fact their null spaces arehuge.

Theorem 2.12. Pf = 0 if and only if f = ∇ψ for some ψ.

The nontrivial part of the proof makes use of Theorem 2.5: If Pf = 0, then,for ξ ∈ θ⊥

(Pf)∧(θ, ξ) = θ · (Pf)∧(θ, ξ) = (2π)1/2θ · f(ξ) = 0 .

Hence θ⊥ is invariant under f . This is only possible if f(ξ) = iψ(ξ)ξ with someψ(ξ). It remains to show that ψ is sufficiently regular to conclude that f = ∇ψ.

A similar proof leads to

Theorem 2.13. Let n = 3. R⊥f = 0 if and only if f = curl φ for some φ.

Inversion formulas can be derived for those parts of the solution that areuniquely determined. We make use of the Helmholtz decomposition to write avector f in the form

f = fs + f i

where fs is the solenoidal part (i.e. div fs = 0 or fs = curl φ) and f i is theirrotational part (i.e. curl f i = 0, f i = ∇ψ). Then, Theorem 2.12 states that thesolenoidal part of f is uniquely determined by Pf , while the irrotational part off is uniquely determined by R⊥f .

We have the following inversion formula:

Theorem 2.14. Let g = Pf and let f be solenoidal (i.e. f i = 0). Then,

f =n − 1

2π|Sn−2| P∗I−1g ,

(P∗g)(x) =∫

Sn−1

g(θ,Eθx)θdθ

and I−1 is the Riesz potential already used in Theorem 2.6.

For more results see Sharafutdinov (1994).

26 F. Natterer

3 Reconstruction Algorithms

There are essentially three classes of reconstruction algorithms. The first classis based on exact or approximate inversion formulas, such as Radon’s inversionformula. The prime example is the filtered backprojection algorithm. It’s not onlythe work horse in clinical radiology, but also the model for many algorithms inother fields. The second class consists of iterative methods, with Kaczmarz’smethod (called algebraic reconstruction technique (ART) in tomography) asprime example. The third class is usually called Fourier methods, even thoughFourier techniques are used also in the first class. Fourier methods are directimplementations of the projection slice theorem (Theorems 2.1, 2.5) without anyreference to integral geometry.

3.1 The Filtered Backprojection Algorithm

This can be viewed as an implementation of Radon’s inversion formula(Theorem 2.2). However it is more convenient to start out from the formula

Theorem 3.1. Let V = R∗v. Then,

V ∗ f = R∗(v ∗ g)

where the convolution on the left hand side is in Rn, i.e.

(V ∗ f)(x) =∫

Rn

V (x − y)f(y)dy

and the convolution on the right hand side is in R1, i.e.

(v ∗ g)(θ, s) =∫

R1

v(θ, s − t)g(θ, t)dt .

The proof of Theorem 3.1 require not much more than the definition of R∗.Theorem 3.1 is turned into an (approximate) inversion formula for R by

choosing V ∼ δ. Obviously V is the result of the reconstruction procedure forf = δ. Hence V is the point spread function, i.e. the function the algorithmwould reconstruct if the true f were the δ-function.

Theorem 3.2. Let V = R∗v with v independent of θ. Then,

V (ξ) = 2(2π)(n−1)/2|ξ|1−nv(|ξ|) .

For V to be close to δ we need V to be close to (2π)−n2 , hence

v(σ) ∼ 12(2π)−n+1/2σn−1 .

Let φ be a low pass filter, i.e. φ(σ) = 0 for |σ| > 1 and, for convenience, φeven. Let Ω be the (spatial) bandwidth. By Shannon’s sampling theorem Ωcorresponds to the (spatial) resolution 2π/Ω:

X-ray Tomography 27

Theorem 3.3. Let f be Ω-bandlimited, i.e. f(ξ) = 0 for |ξ| > Ω. Then, f isuniquely determined by f(h), ∈ Z

n, h ≤ π/Ω. Moreover, if h ≤ 2π/Ω, then∫

Rn

f(x)dx = hn∑

f(h) .

An example for a Ω-bandlimited function in R1 is f(x) = sinc(Ωx) where

sinc(x) = sin(x)/x is the so-called sinc-function. From looking at the graph of fwe conclude that bandwidth Ω corresponds to resolution 2π/Ω.

We remark that Theorem 3.3 has the following corollary:Assume that f , g are Ω-bandlimited. Then,

∫

Rn

f(x)g(x)dx = hn∑

f(h)g(h)

provided that h ≤ π/Ω. This is true since fg has bandwidth 2Ω.We put

v(σ) = 2(2π)(n−1)/2|σ|n−1φ(σ/Ω) .

v is called the reconstruction filter. As an example we put n = 2 and take as φthe ideal low pass, i.e.

φ(σ) =

⎧⎨⎩

1 , |σ| ≤ 1 ,

0 , |σ| > 1 .

Then,

v(s) =Ω2

4π2u(Ωs) , u(s) = sinc(s) − 1

2

(sinc

(s

2

))2

.

Now we describe the filtered backprojection algorithm for the reconstruction ofthe function f from g = Rf . We assume that f has the following properties:

f is supported in |x| ≤ ρ (13)

f is essentially Ω-bandlimited, i.e. (14)

f(ξ) is negligible in some sense for |ξ| > Ω .

Since f is essentially Ω-bandlimited we conclude from Theorem 2.1 that thesame is true for g. Hence, by a loose application of the sampling theorem,

(v ∗ g)(θ, s) = ∆s∑

k

v(θ, s − sk)g(θ, sk) (15)

where s = ∆s and ∆s ≤ π/Ω. Equation (15) is the filtering step in the filteredbackprojection algorithm. In the backprojection step we compute R∗(v ∗ g) in adiscrete setting. We restrict ourselves to the case n = 2, i.e.

28 F. Natterer

(R∗(v ∗ g))(x) =∫

S1

(v ∗ g)(θ, x·)dθ .

In order to discretize this integral properly we make use of Theorem 3.3 again.For this we have to determine the bandwidth of the function ϕ → (v ∗ g)(θ, x · θ)

where θ =

⎛⎝ cos ϕ

sin ϕ

⎞⎠ .

Theorem 3.4. For k an integer we have

2π∫

0

e−ikϕ(v ∗ g)(θ, x · θ)dϕ =ik

2π

∫

|y|<ρ

f(y)e−ikψ

Ω∫

−Ω

vk(σ)Jk(−σ|x − y|)dσdy

where ψ = arg(y) and Jk is the Bessel function of order k of the first kind.

All that is needed for the proof is the integral representation

Jk(s) =i−k

2π

2π∫

0

e−ikϕ+s cos ϕdϕ

of the Bessel function.By Debye’s asymptotic relation (see Abramowitz and Stegun (1970)), Jk(s) is

negligibly small for |s| < |k| and |k| large. Hence the left hand side in Theorem 3.4is negligible for 2Ωρ < |k|. In other words, the function ϕ → (v ∗ g)(θ, x · θ) hasessential bandwidth 2Ωρ. According to Theorem 3.3,

(R∗(v ∗ g))(x) = ∆ϕ∑

j

(v ∗ g)(θj , x · θj) (16)

where θj =

⎛⎝ cos ϕj

sin ϕj

⎞⎠, ϕj = j∆ϕ, provided that ∆ϕ ≤ π/Ωρ.

The formulas (16), (17) describe the filtered backprojection algorithm – ex-cept for an interpolation step in the second argument. It is generally acknowl-edged that linear interpolation suffices.

3.2 Iterative Algorithms

The equations one has to solve in CT are usually of the form

Rjf = gj , j = 1, . . . , p (17)

X-ray Tomography 29

where Rj are certain linear operators from a Hilbert space into an Hilbert spaceHj . For instance, for the 2D problem we have

(Rjf)(s) = (Rf)(θj , s) . (18)

Kaczmarz’s method solves linear systems of the kind (18) by successively pro-jecting orthogonally onto the affine subspaces Rjf = gj of H:

fj = Pjmodpfj−1 , j = 1, 2, . . . .

Here, Pj is the orthogonal projection onto Rjf = gj , i.e.

Pjf = f − R∗j (RjR

∗j )

−1(Rjf − gj) .

For the application to tomography we replace the operator RjR∗j by a simpler

one, Cj , and introduce a relaxation parameter ω. Putting

Pjf = f − R∗jC

−1j (Rjf − gj) ,

Pωj f = (1 − ω)f + ωPjf

we have the following convergence result.

Theorem 3.5. Let (17) be consistent. Assume that Cj ≤ RjR∗j (i.e. Cj her-

mitian and Cj − RjR∗j positive semidefinite) and 0 < ω < 2. Then, fj =

Pωjmodpfj−1 converges for f0 = 0 to the solution of (18) with minimal norm.

The condition 0 < ω < 2 is reminiscent of the SOR (successive overre-laxation) method of numerical analysis. In fact Kaczmarz’s method can beviewed as an SOR method applied to the linear system R∗Rf = R∗g whereR = R1 × · · · × Rp.

Theorem 3.5 can be applied directly to tomographic problems. The conver-gence behaviour depends critically on the choice of ω and of the ordering of (17).We analyse this for the standard case (18). For convenience we assume that f issupported in |x| < 1 and gj = 0, j = 1, . . . , p.

The following result is due to Hamaker and Solmon (1978):

Theorem 3.6. Let Cm be the linear subspace of L2(|x| < 1) spanned by Tm(x ·θ1), . . . , Tm(x · θp), Tm the first kind Chebysheff polynomials. Then,

PjCm ⊆ Cm .

This means that Cm is an invariant subspace under the iteration ofTheorem 3.6. Thus we can study the convergence behaviour on each of thesefinite dimensional subspaces separately. Let ρm(ω) be the spectral radius of

Pω = Pω1 · · ·Pω

p .

ρm(ω) can be computed numerically:

30 F. Natterer

0

1

0 30

ω = 0.1

ω = 0.5

ω = 1.0

m

ρm(ω)

0

1

0 30

ω = 0.1

ω = 0.5

ω = 1.0

m

ρm(ω)

0

1

0 30

ω = 0.1

ω = 0.5

ω = 1.0

m

ρm(ω)

First we consider the solid lines. They correspond to ω = 1. For the sequentialordering ϕj = jπ/p, j = 1, . . . , p the values of ρm(ω) are displayed in the topfigure. We see that ρm(ω) is fairly large for low order subspaces Cm, indicatingslow convergence for low frequency parts of f , i.e. for the overall features of f .The other figures shows the corresponding values of ρm(ω) for a non-consecutiveordering of the ϕj suggested by Herman and Meyer (1993) and for a randomordering respectively. Both orderings yield much smaller values of ρm(ω), in

X-ray Tomography 31

particular on the low order subspaces. The conclusion is that for practical workone should never use the consecutive ordering, and random ordering is usuallyas good as any other more sophisticated ordering.

In particular in emission tomography the use of multiplicative iterative meth-ods is quite common. The most popular of these multiplicative algorithms is theexpectation-maximization (EM) algorithm: Let Af = g be a linear system, withall the elements of the (not necessarily square) matrix A being non-negative.Assume that A is normalized such that A1 = 1 where 1 stands for vectors ofsuitable dimension containing only 1′s. The EM algorithm for the solution ofAf = g reads

f j+1 = f jA∗ g

Af j, j = 1, 2, . . . ,

where division and multiplication are understood componentwise. One can showthat f j converges to the minimizer of the log likelihood function

(f) =∑

i

(gi log(Af)i − (Af)i) .

There exists also a version of the EM algorithm that decomposes the systemAf = g into smaller ones, Ajf = gj , j = 1, . . . , p, as in the Kaczmarz method(ordered subset EM, OSEM; see Hudson and Larkin (1992)). As in the Kaczmarzmethod one can obtain good convergence properties by a good choice of theordering, and the convergence analysis is very much the same.

For more on iterative methods in CT see Censor (1981).

3.3 Fourier Methods

Fourier methods make direct use of relations such as Theorem 2.1. Let’s considerthe 2D case in which we have

f(σθ) = (2π)−1/2g(θ, σ) , g = Rf . (19)

In order to implement this we first have to do a 1D discrete Fourier transformon g for each direction θj :

g(θj , σ) = (2π)−1/2∆s∑

k

e−iσskg(θj , sk)

where sk = k∆s and σ = ∆σ. This provides f at the points σθj whichform a grid in polar coordinates. The problem is now to do a 2D inverse Fouriertransform on this polar coordinate grid. Various methods have been developed fordoing this, in particular the gridding method and various fast Fourier transformon non-equispaced grids; see Fourmont (1999), Beylkin (1995), Steidl (1998).

32 F. Natterer

4 Reconstruction from Cone Beam Data

One of the most urgent needs of present day’s X-ray CT is the developmentof accurate and efficient algorithms for the reconstruction of f from (Cf)(a, θ)where a is on a source curve A (typically a helix) and θ ∈ S2. In practice, θ isrestricted to a subset of S0, but we ignore this additional difficulty.

We assume that the source curve A satisfies Tuy’s condition, i.e. A intersectseach plane hitting supp (f) transversally. Let A be the curve x = a(λ), λ ∈ Λ.Tuy’s condition implies that for each s with (Rf)(θ, s) = 0 there is λ such thats = a(λ) · θ. In fact there may be more than one such λ, but we assume thattheir number is finite. Then we may choose a function M(θ, s) such that

∑λ

s=a(λ)·θ

M(θ, λ) = 1

for each s. Let g(λ, θ) = (Cf)(a(λ), θ) be the data function. Put

G(λ, θ) =∫

θ⊥∩S2

∂

∂θg(λ, ω)dω .

Then we have

Theorem 4.1. Let w be a function in R1 and V = R∗w′. Then,

(V ∗ f)(x) =∫

S2

∫

Λ

w((x − a(λ)) · θ)G(λ, θ)|a′(λ) · θ|M(θ, λ)dλdθ .

For the proof we start out from the 3D case of Theorem 3.1:

(V ∗ f)(x) =∫

S2

∫

R1

w′(x · θ − s)(Rf)(θ, s)dsdθ

=∫

S2

∫

R1

w(x · θ − s)(Rf)′(θ, s)dsdθ .

In the s integral we make the substitution s = a(λ) · θ, obtaining

(V ∗ f)(x) =∫

S2

∫

Λ

w((x − a) · θ)(Rf)′(θ, a(λ) · θ)|a′(λ) · θ|M(θ, λ)dλdθ ,

the factor M(θ, s) being due to the fact that λ → a(λ) · θ is not one-to-one. ByTheorem 2.8,

(Rf)′(θ, a(λ) · θ) = G(λ, θ) ,

and this finishes the proof.

X-ray Tomography 33

Theorem 4.1 is the starting point for a filtered backprojection algorithm. For

w′ = − 18π2

δ′ ,

δ the 1D δ-function, we have V = δ, the 3D δ-function, hence

f(x) =∫

Λ

|x − a(λ)|−2Gw

(λ,

x − a(λ)|x − a(λ)|

)dλ ,

Gw(λ, ω) = − 18π2

∫

S2

δ′(ω · θ)G(λ, θ)|a′(λ) · θ|M(θ, λ)dθ, ω ∈ S2 .

This is the formula of Clack and Defrise (1994) and Kudo and Saito (1994). Formore recent work see Katsevich (2002).

There exist various approximate solutions to the cone-beam reconstructionproblem, most prominently the FDK formula for a circular source curve (whichdoesn’t suffice Tuy’s condition). It can be viewed as a clever adaption of the 2Dfiltered backprojection algorithm to 3D; see Feldkamp, Davis and Kress (1984).An approximate algorithm based on Orlov’s condition of Theorem 2.7 is the πmethod of Danielsson et al. (1999).

References

1. M. Abramowitz and I.A. Stegun (1970): Handbook of Mathematical Functions.Dover.

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3. Y. Censor (1981): ‘Row–action methods for huge and sparce systems and theirapplications’, SIAM Review 23, 444-466.

4. R. Clack and M. Defrise (1994): ‘A Cone–Beam Reconstruction Algorithm UsingShift–Variant Filtering and Cone–Beam Backprojection’, IEEE Trans. Med. Imag.13, 186-195.

5. P.E. Danielsson, P. Edholm, J. Eriksson, M. Magnusson Seger, and H. Turbell(1999): The original π-method for helical cone-beam CT, Proc. Int. Meeting onFully 3D-reconstruction, Egmond aan Zee, June 3-6.

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34 F. Natterer

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IEEE Press, New York.17. A. Katsevich (2002): ‘Theoretically exact filtered backprojection-type inversion

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